Comparison optimal allocation solutions

rm(list = ls())
source("C:/Users/mbofi/Dropbox/CeMSIIS/GitHub/Allocation/case-study/aux_functions.R") #local
# setwd("C:/Users/mbofi/Dropbox/CeMSIIS/GitHub/Allocation/case-study")#local
# setwd("~/GitHub/Allocation/case-study")

library(tidyverse)
library(dplyr)
library(plyr)
library(NCC)
library(mmtable2)
library(gt)

# OPTIMAL ALLOCATION USING OLD NOTATION
##########################################
# Simulation function
# modification of sim_designs() function
# also returning the matrix (r_is)
##########################################

sim_designs_rr <- function(r1,r2,mu0,mu1,mu2,N,alloc="sqrt",trend="stepwise",sl=0.2){
  
  r3 = 1-r1-r2
  
  if(r1 == 1){
    
    if(alloc == "one"){
      r22 <- r11 <- r01 <- r1/3
      r02 <- r12 <- r23 <- r03 <- 0
    }
    if(alloc != "one"){ 
      r01 <- sqrt(2)/(2+sqrt(2))
      r22 <- r11 <- (1-r01)/2 
      r02 <- r12 <- r23 <- r03 <- 0
    }
    
  }else{
    
    if(alloc == "opt"){ 
      
      r11 <- r01 <- r1/2
      
      r2_opt <- f(r1=r1,r2=r2)
      r02 <- r2_opt[3]
      r12 <- r2_opt[4]
      r22 <- r2_opt[5]
      
      r23 <- r03 <- r3/2
      
    }
    if(alloc == "one"){
      r11 <- r01 <- r1/2
      r22 <- r12 <- r02 <- r2/3
      r23 <- r03 <- r3/2
    }
    if(alloc == "sqrt"){
      r11 <- r01 <- r1/2
      r02 <- r2*sqrt(2)/(2+sqrt(2))
      r22 <- r12 <- (r2-r02)/2
      r23 <- r03 <- r3/2
    }
    
  }
  # c(r11,r01)
  
  n11 = round(r11*N)
  n01 = round(r01*N)
  
  n22 = round(r22*N)
  n12 = round(r12*N)
  n02 = round(r02*N)
  
  n23 = round(r23*N)
  n03 = round(r03*N)
  
  # c(r11,r01,r22,r12,r02,r23,r03)
  # c(n11,n01,n22,n12,n02,n23,n03)
  
  means = c(mu0,mu1,mu2)
  
  treatment = c(
    sample(c(rep(1,n11),rep(0,n01))),
    sample(c(rep(2,n22),rep(1,n12),rep(0,n02))),
    sample(c(rep(2,n23),rep(0,n03)))
  )
  
  Nsim = length(treatment) 
  
  if(r1==1){
    treatment = sample(treatment)
    period = rep(1,Nsim)
  }else{
    period = c(
      rep(1,n11+n01),
      rep(2,n22+n12+n02),
      rep(3,n23+n03)
    )
  }
  
  if(trend=="stepwise"){
    
    response = rnorm(Nsim,
                     mean=means[treatment[1:Nsim]+1]+sw_trend(cj=period[1:Nsim], lambda=sl),
                     sd=1) 
    
  }
  if(trend=="linear"){
    
    response = rnorm(Nsim,
                     mean=means[treatment[1:Nsim]+1]+linear_trend(j=1:Nsim, lambda=sl, sample_size=c(0,Nsim)),
                     sd=1) 
  }
  
  data = data.frame(response,treatment,period)
  if(r1==1){
    ss = matrix(c(n22,n11,n01,0,n12,n02,n23,0,n03), nrow=3)
  }else{
    ss = matrix(c(0,n11,n01,n22,n12,n02,n23,0,n03), nrow=3)
  }
  r_matrix = matrix(c(0,r11,r01,r22,r12,r02,r23,0,r03), nrow=3)
  r_periods = c(r1,r2,r3)
  return(list(data=data,ss=ss,r_matrix=r_matrix,r_periods=r_periods))
}

# OPTIMAL ALLOCATION USING THE NEW NOTATION
##########################################
eq_p22 <- function(p22,r1=0.1,r2=0.8){(Power(-1 + p22,3)*(-1 + 2*r1))/((-1 + 2*p22)*(-2 + p22*(7 + p22*(-15 + p22*(19 + 2*p22*(-7 + 2*p22))))))-r2}

f_p=Vectorize(function(r1,r2) { 
  p22=uniroot(eq_p22,c(0,r2),r1=r1,r2=r2)$root
  r22 = r2*p22
  # p02 <- 1/(2-2*p22)
  p02 <- 1/(2-2*p22)-p22
  r02 <- p02*r2
  
  p12 <- 1-p02-p22
  r12 <- p12*r2
  # r12 <- r2- r02- r22
  
  sol=c(r1,r2,r02,r12,r22)
  sol
})

Solutions case study

Design 3: three-period design (symmetric design)

##########################################
# Case study
##########################################

# original study - obtained means
mean_control = 17.3/3.5
mean_arm1 = 66.2/3.5
mean_arm2 = 72.3/3.5 

##########################################
# design 3: three-period design (symmetric design)
##########################################
N = 92
N1 = round(N/3)
N3 = round(N/3)
N2 = N-N1-N3
c(N1,N2,N-N1-N2)

[1] 31 30 31

alloc_str <- c("one","opt","sqrt")

Optimal allocation using the old parametrization

m <- sim_designs_rr(r1=N1/N,r2=N2/N,
                   mu0=mean_control,mu1=6,mu2=6,
                   N=N,alloc=alloc_str[2],sl=0)
  
  rownames(m$r_matrix)  <- c("Arm 2", "Arm 1", "Control")
  knitr::kable(m$r_matrix, format = "html", caption = paste(alloc_str[2]), 
               col.names = c("Period 1", "Period 2", "Period 3"), 
               row.names=T, 
               digits=3)
  
  knitr::kable(t(m$r_periods),col.names = c("Period 1", "Period 2", "Period 3"))

Optimal allocation using the new parametrization

rownames(as.matrix(r_solutions[3:5]))  <- c("Arm 2", "Arm 1", "Control")

Error in rownames(as.matrix(r_solutions[3:5])) <- c("Arm 2", "Arm 1",  : 
  could not find function "as.matrix<-"

Design 3: three-period design (non-symmetric design)

##########################################
# design 3: three-period design (non-symmetric design)
##########################################
N = 92
N1 = round(N/3)
N2 = round(2*(N-N1)/3)
c(N1,N2,N-N1-N2)

Optimal allocation using the old parametrization

m <- sim_designs_rr(r1=N1/N,r2=N2/N,
                   mu0=mean_control,mu1=6,mu2=6,
                   N=N,alloc=alloc_str[2],sl=0)
  
  rownames(m$r_matrix)  <- c("Arm 2", "Arm 1", "Control")
  knitr::kable(m$r_matrix, format = "html", caption = paste(alloc_str[2]), 
               col.names = c("Period 1", "Period 2", "Period 3"), 
               row.names=T, 
               digits=3)
  
  knitr::kable(t(m$r_periods),col.names = c("Period 1", "Period 2", "Period 3"),digits=3)

Optimal allocation using the new parametrization

r_solutions

              [,1]
r1      0.33695652
r2      0.32608696
Arm 2   0.13506964
Arm 1   0.09550527
Control 0.09551205